The main result of this paper is that any weak solution \(u\)  of (REF ) is Hölder continuous in the interior of \(\Omega \times (0,T]\) , provided that \(u\)  is bounded away from 1.
This builds upon our earlier work [1]}, where we showed the well-posedness of (REF ) and (REF ) on bounded Lipschitz domains for initial and boundary value problems with mixed boundary conditions.
Indeed, we infer from our main result that the solutions of (REF ) obtained in [1]} are Hölder continuous in the interior of the domain.
The same holds for the first equation of (REF ) implying interior Hölder continuity for \(M\) .
Moreover, the second equation of (REF ) is non-degenerate so classical results on interior Hölder continuity can be applied to obtain interior Hölder continuity for \(C\) , see for example [3]}.
